Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

TL;DR

This paper develops new Carleman estimates for variable coefficient degenerate elliptic operators, extending previous results and applying them to establish unique continuation properties and quantitative uniqueness results, including for Hardy potentials.

Contribution

It introduces generalized Carleman estimates for variable coefficient degenerate elliptic operators, broadening the scope of previous constant coefficient results and enabling new unique continuation theorems.

Findings

Established Bourgain-Kenig type quantitative uniqueness in variable coefficient setting

Proved strong unique continuation for degenerate sublinear equations

Derived a subelliptic Carleman estimate leading to unique continuation with Hardy potentials

Abstract

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.